Kepler extracted from Tycho Brahé’s punctilious measurements the three great planetary laws which we now call by his name. These experiments provide ways of explaining the vital second law. Practical Activity for 14-16
Class practical A simple method for drawing an ellipse, using the two foci. Apparatus and Materials For each student or group of students
Health & Safety and Technical Notes Read our standard health & safety guidance Procedure
Teaching Notes
Practical Activity for 14-16
Demonstration Three methods of visualising elliptical orbits. Apparatus and Materials Either Method 1:
Or Method 2:
Or Method 3:
Health & Safety and Technical Notes Read our standard health & safety guidance For method 2, stretch the rubber sheeting over the rigid horizontal circular frame and secure it tightly with tape. It should be stretched a little, equally in all directions. Fix a vertical metal rod over the sheet so that it is pushing the sheet into a curved well, thus imitating, roughly, an inverse-square-force potential. For method 3, the apparatus is available from a number of suppliers: Philip Harris, Griffin Education, or ASCOL. Procedure Method 1: Firmly hold the glass funnel vertically and let the ball fall into it. Friction will affect the orbit and make it precess, but the motion around the funnel will be elliptic. Select a ball which will fall right through the funnel. Method 2: Project the small steel ball across the sheet. By choosing suitable initial conditions the ball can be made to describe an oval like an ellipse with one focus on the axis of the well. Method 3: Balance the plastic or aluminium hill upside down. Wood blocks can be used to hold it in position. Project the ball across the inverted hill so that it will orbit the centre. The elliptical path will be visible. Teaching Notes
Practical Activity for 14-16
Demonstration Demonstrating the law using air or carbon dioxide pucks – or an ice rink. Apparatus and Materials
Health & Safety and Technical Notes Take care when handling the glass plate. If taking a class off site, follow your employer's procedures for off-site educational visits. Read our standard health & safety guidance Procedure
Teaching Notes
Practical Activity for 14-16
This simple class activity demonstrates the increase in speed with a diminishing radius of orbit. Apparatus and Materials For each student or group of students
Health & Safety and Technical Notes Read our standard health & safety guidance Procedure
Teaching Notes
Practical Activity for 14-16
Demonstration The speed of the puck increases as the radius of orbit of the puck decreases. Apparatus and Materials Health & Safety and Technical Notes Dry ice is very cold. Wear thermal gloves to handle it, and wear safety spectacles. Take care when handling the glass plate. Read our standard health & safety guidance Procedure
Teaching Notes As the force on the puck is increased, its orbit gets smaller and its speed increases. Kepler's Second Law states that a planet in its orbit sweeps out equal areas in equal times.
Practical Activity for 14-16
Class practical The simple kit demonstrates the increase in speed with a diminishing radius of orbit. Apparatus and Materials For each student or group of students Health & Safety and Technical Notes Ensure that the bungs are securely attached and that each student has sufficient space to whirl the bung safely. The centripetal forces kit is available from: Technology Supplies Limited Procedure
Teaching Notes
Practical Activity for 14-16
Demonstration Apparatus and Materials
Health & Safety and Technical Notes Supervise students so they don't spin too quickly, or they may topple over. A freely spinning music stool (or office chair) may be used for these demonstrations. It should be oiled before use. Procedure
Teaching Notes This experiment was safety-tested in March 2008
Teaching Guidance for 14-16
When Tycho Brahe was 17, he observed the conjunction of Jupiter and Saturn and was dismayed to find that the astronomical tables of the time were inaccurate in predicting the event by as much as a month. He decided to devote his life to making better tables, for which purpose he constructed better and better instruments. The birth of modern planetary astronomy, with the three planetary laws discovered by Kepler, was based on the precise observations resulting from Tycho Brahe’s passion for accuracy. Kepler: Law-giver of the heavensIn the course of his lifetime, Kepler extracted the three great planetary laws which we now call by his name. The third law, which binds the movements of the planets together mathematically, Kepler discovered, with tremendous delight, quite late in life. Mapping the Earth’s orbit in space and timeTo map the Earth’s orbit around the Sun on a scale diagram you need many sets of measurements, each set giving the Earth’s bearings from two fixed points. Kepler took the fixed Sun for one of these and for the other he took Mars at a series of times when it was in the same position in its orbit. Kepler proceeded thus: he marked the ‘position’ of Mars in the star pattern at one position (opposite the Sun, overhead at midnight). That gave him the direction of a base line, Sun – Earth – Mars, SE1M. Then he turned the pages of Tycho’s records to a time exactly one Martian year later. (The time of Mars’ motion around its orbit was known accurately from records over many centuries). Keplers Scheme to plot the Earths orbit. Then Kepler knew that Mars was in the same position, M, so that SM had the same direction. By now, the Earth had moved on to E2 in its orbit. Tycho’s record of the position of Mars in the star pattern gave him the new apparent direction of Mars E2 M and the Sun’s position gave him E2 S. Then he could calculate the angles of the triangle SE2M from the record thus: since he knew the directions E1 M and E2 M (marked on the celestial sphere of stars) he could calculate angle A between them. Since he knew the directions E1 S and E2 S he could calculate angle B. Then on a scale diagram he could choose two points to represent S and M and locate the Earth’s position,E2 as follows. At the ends of the fixed base line SM, draw lines making angles A and B and mark their intersection E2 . One Martian year later he could find the directions E3 M and E3 S from the records and mark E3 on his diagram. Thus Kepler could start with the points S and M and locate E2 ,E3 ,E4 ..... enough points to show the orbit’s shape. Knowing the Earth’s true orbit he could invert the investigation and plot the shape of Mars’ orbit. He found that he could treat the Earth’s orbit either as an eccentric circle or as slightly oval but Mars’ orbit was far from circular: it was definitely oval. It was an ellipse with the Sun at one focus – Kepler’s First Law of planetary motion. Planetary data and Kepler’s Third LawKepler continued to brood on one of his early questions: what connection is there between the size of the planet’s orbit and the times of its year? Students can try and investigate the relationship between the planetary orbit radius, R , and the orbital time, T, using modern data. These are more accurate than the data available to Kepler. It will become obvious, fairly quickly, that simple proportion will not do. For example as R almost doubles in going from Mercury to Venus, T, almost triples; as R grows almost 10 times from Earth to Saturn, T , grows about 30 times. Kepler wrestled with this for a very long time, trying different combinations, until he found that R 3T 2 was a constant. Kepler was overjoyed! His three laws were clear, simple and powerful and they fitted the facts very accurately. He earned the title law-giver of the heavens.
Early astronomers, in different civilizations, used the observed motion of the stars, the Sun, Moon and planets as the basis for clocks, calendars and a navigational compass. The Greeks developed models to account for these celestial motions. Copernicus, in the 16th century, was the first to explain the observed looping (retrograde) motion of planets, by replacing a geocentric heliocentric model of the Universe with a heliocentric model. Modern planetary astronomy really began in the 17th century with Kepler, who used Tycho Brahe’s very accurate measurements of the planetary positions to develop his three laws. Galileo contributed to the development of astronomy by teaching the Copernican view, and by devising a telescope which he used to show Jupiter’s moons as a model for the solar system, among other things. Newton built on earlier insights with his universal law of gravitation and its fruits: predictions or explanations of Kepler’s laws, the motion of comets, the shape of the Earth, tides, precession of the equinoxes and perturbations in the motion of planets which led to the discovery of Neptune. He also had to invent the mathematics to do this: calculus. |